Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

Isaac Azure

doi:doi:10.11648/j.ajam.20241204.11

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Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

Isaac Azure^*

Published in American Journal of Applied Mathematics (Volume 12, Issue 4)

Received: 2 June 2024 Accepted: 17 June 2024 Published: 6 August 2024

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Abstract

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (MT_3) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 4)
DOI	10.11648/j.ajam.20241204.11
Page(s)	79-90
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Newton-Raphson Method, Broyden Method, Quadrature Rules, Composite Trapezoidal Rule

References

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[3]	Al-Towaiq, M. H., & Abu Hour, Y. S. (2017). Two improved classes of Broyden's methods for solving nonlinear systems of equations. JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE-JMCS, 17(1), 22-31.
[4]	Kou, J., Li, Y., & Wang, X. (2007). A composite fourth-order iterative method for solving non-linear equations. Applied Mathematics and Computation, 184(2), 471-475.
[5]	Luo, Y. Z., Tang, G. J., & Zhou, L. N. (2008). Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing, 8(2), 1068-1073.
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[7]	Osinuga, I. A., & Yusuff, S. O. (2017). Construction of a Broyden-like method for Nonlinear systems of equations. Annals. Computer Science Series, 15(2), 128-135.
[8]	Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton's method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93.
[9]	Frontini, M. A. R. C. O., & Sormani, E. (2003). Some variant of Newton’s method with third-order convergence. Applied Mathematics and Computation, 140(2-3), 419-426.
[10]	Dhamacharoen, A. (2014). An efficient hybrid method for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 263, 59-68.
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[12]	Mohammad, H., & Waziri, M. Y. (2015). On Broyden-like update via some quadratures for solving nonlinear systems of equations. Turkish Journal of Mathematics, 39(3), 335-345. https://doi.org/10.3906/mat-1404-41
[13]	Isaac, A., Stephen, T. B., Seidu, B., Maza, D. D. A., Olukotun, S. F., & Akinlade, G. O. (2021). A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. American Journal of Mathematical and Computer Modelling, 6(1), 1-8. https://doi.org/10.11648/J.AJMCM.20210601.11
[14]	Osinuga, I. A., & Yusuff, S. O. (2018). Quadrature based Broyden-like method for systems of nonlinear equations. Statistics, Optimization & Information Computing, 6(1), 130-138. https://doi.org/10.19139/soic.v6i1.471
[15]	Darvishi, M. T., & Shin, B. C. (2011). High-order Newton-Krylov methods to solve systems of nonlinear equations. Journal of the Korean Society for Industrial and Applied Mathematics, 15(1), 19-30.
[16]	Hafiz, M. A., & Bahgat, M. S. (2012). An efficient two-step iterative method for solving system of nonlinear equations. Journal of Mathematics Research, 4(4), 28. https://doi.org/10.5539/jmr.v4n4p28
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[21]	Isaac, A., Stephen, T. B., & Seidu, B. (2021). A Comparison of Newly Developed Broyden–Like Methods for Solving System of Nonlinear Equations. International Journal of Systems Science and Applied Mathematics, 6(3), 77-94. https://doi.org/10.11648/j.ijssam.20210603.11

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APA Style

Azure, I. (2024). Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. American Journal of Applied Mathematics, 12(4), 79-90. https://doi.org/10.11648/j.ajam.20241204.11

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ACS Style

Azure, I. Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. Am. J. Appl. Math. 2024, 12(4), 79-90. doi: 10.11648/j.ajam.20241204.11

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AMA Style

Azure I. Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. Am J Appl Math. 2024;12(4):79-90. doi: 10.11648/j.ajam.20241204.11

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@article{10.11648/j.ajam.20241204.11,
  author = {Isaac Azure},
  title = {Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {4},
  pages = {79-90},
  doi = {10.11648/j.ajam.20241204.11},
  url = {https://doi.org/10.11648/j.ajam.20241204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241204.11},
  abstract = {The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (MT_3) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.
},
 year = {2024}
}

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TY  - JOUR
T1  - Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

AU  - Isaac Azure
Y1  - 2024/08/06
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241204.11
DO  - 10.11648/j.ajam.20241204.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 79
EP  - 90
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241204.11
AB  - The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (MT_3) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

VL  - 12
IS  - 4
ER  -

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Author Information

Isaac Azure

Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Contact Email

http://orcid.org/0009-0008-0861-541X

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APA Style

Azure, I. (2024). Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. American Journal of Applied Mathematics, 12(4), 79-90. https://doi.org/10.11648/j.ajam.20241204.11

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ACS Style

Azure, I. Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. Am. J. Appl. Math. 2024, 12(4), 79-90. doi: 10.11648/j.ajam.20241204.11

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AMA Style

Azure I. Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods. Am J Appl Math. 2024;12(4):79-90. doi: 10.11648/j.ajam.20241204.11

Copy | Download

@article{10.11648/j.ajam.20241204.11,
  author = {Isaac Azure},
  title = {Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {4},
  pages = {79-90},
  doi = {10.11648/j.ajam.20241204.11},
  url = {https://doi.org/10.11648/j.ajam.20241204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241204.11},
  abstract = {The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (MT_3) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

AU  - Isaac Azure
Y1  - 2024/08/06
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241204.11
DO  - 10.11648/j.ajam.20241204.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 79
EP  - 90
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241204.11
AB  - The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (MT_3) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

VL  - 12
IS  - 4
ER  -

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